Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}6x-7y &= -9 \\ -9x+4y &= -6\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-9x = -4y-6$ Divide both sides by $-9$ to isolate $x$ $x = {\dfrac{4}{9}y + \dfrac{2}{3}}$ Substitute this expression for $x$ in the first equation. $6({\dfrac{4}{9}y + \dfrac{2}{3}}) - 7y = -9$ $\dfrac{8}{3}y + 4 - 7y = -9$ Simplify by combining terms, then solve for $y$ $-\dfrac{13}{3}y + 4 = -9$ $-\dfrac{13}{3}y = -13$ $y = 3$ Substitute $3$ for $y$ in the top equation. $6x-7( 3) = -9$ $6x-21 = -9$ $6x = 12$ $x = 2$ The solution is $\enspace x = 2, \enspace y = 3$.